Locally Finite Set of Subsets is Sigma-Locally Finite Set of Subsets

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Theorem

Let $T = \struct {S, \tau}$ be a topological space.

Let $\AA$ be a locally finite set of subsets.

Then:

$\AA$ is a $\sigma$-locally finite set of subsets

Proof

For each $n \in \N$, let

$\AA_n = \AA$.

Then:

$\AA = \ds \bigcup_{n \in \N} \AA_n$

The result follows.

$\blacksquare$